Uncertain minimum cost multicommodity flow problem

Abstract

This paper investigates the minimum cost multicommodity flow problem with uncertain costs and uncertain capacities. Uncertainty theory is used to deal with indeterminacy factors in uncertain network. An ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow model is formulated. Some properties of the model are analyzed. An equivalence relationship between the ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow and the minimum cost multicommodity flow is proved. Then, a global optimal algorithm is developed for finding the ($$\alpha $$ź, $$\beta $$β)-minimum cost multicommodity flow and the minimum cost multicommodity flow, and obtaining the uncertainty distribution of total shipping cost. Finally, two examples are presented to illustrate the algorithm.